Mathematical Induction

Q: A farmer raises chickens in a farm. Taking into account various conditions, such as the sale of poultry, deaths, etc that affect the population of chickens in his farm, the number of chickens at the end of n months is modelled by Un, where

Un = 0.875Un-1 + 50

Express U1, U2 and U3 in terms of U0.

Show that Un = (0.875^n)(U0 - 400) + 400 and deduce the value of Un when n becomes very large.

Q: A farmer raises chickens in a farm. Taking into account various conditions, such as the sale of poultry, deaths, etc that affect the population of chickens in his farm, the number of chickens at the end of n months is modelled by Un, where

Un = 0.875Un-1 + 50

Express U1, U2 and U3 in terms of U0.

Show that Un = (0.875^n)(U0 - 400) + 400 and deduce the value of Un when n becomes very large.

Thank you in advance!!

Have you shown that this is true for a base case? will do.

Now assume it true for some , and consider:

but by supposition:

so:

So from our assumption that the result was true for we have proven it true for , hence with the base case this proves the result of all